Typing into Terminal works, AppleScript partially works

What can I do to make my AppleScript run things just like they run in Terminal?

Well, you could tell the Terminal app to run your command:

tell application "Terminal" to do script "youtube-dl  " & input

But, I don't recommend this route, because it will actually open a Terminal window!

There are a whole host of differences between do shell script and typing commands into the Terminal. For starters, do shell script commands are run via sh, rather than the more advanced bash or zsh (And, your "#!/bin/zsh" shabang line at the top will have no effect in an Applescript!).

As to your actual problem, it sounds like youtube-dl can't find your copy of ffmpeg—probably because another difference between the Terminal app and do shell script is your PATH is set differently. Try specifying the location of your ffmpeg binary with youtube-dl's --ffmpeg-location flag.

If it helps to have a reference to look at, here is an Applescript I use to download videos from my web browser with youtube-dl. Note that I've been tweaking this for many years, so it has accumulated a lot of error checking, and code for compatibility with different browsers. Don't be intimidated! https://gist.github.com/Wowfunhappy/c2c8fc097a8b4f47430811fc1d0da041

osascript

#!/usr/bin/osascript
on open location input
    do shell script "/usr/local/bin/youtube-dl  " & quoted form of input
end open location

If run as a script, the hash-bang of your script should be the path to osascript. This is the process that runs AppleScript (an Open Scripting Architecture language) on macOS.

Quote the URL passed into youtube-dl to ensure it appears as a single argument to the tool.

Use the full path to youtube-dl to ensure the tool is found. Use which youtube-dl in Terminal to find the path.

Command Line Arguments

If the AppleScript is being passed the URL as the first command line argument, use the argv handler:

#!/usr/bin/osascript
on run argv
   do shell shell script "/usr/local/bin/youtube-dl " & quoted form of (item 1 of argv)
end run

If this is case, consider removing the script entirely and passing the argument directly to youtube-dl.

Post Processing

youtube-dl has the ability to post-process downloads. You can control this using a youtube-dl configuration file.

To specify the ffmpeg binary to be used add the following line to either a file at /etc/youtube-dl.conf or ~/.config/youtube-dl/config:

--ffmpeg-location /usr/local/bin/ffmpeg

This looks an awful lot like a path problem - you have a script youtube-dl and the program /usr/local/bin/youtube-dl - in your AppleScript, try giving it the full path to your script.